Sugar was traditionally produced and sold as sugarloaves, which are cones of sugar wrapped in paper. Find the total surface area of a cone-shaped sugarloaf with a base diameter of 8 centimeters and a slant height of 22 centimeters. Round answer to the nearest hundredth

Answer:331.09cm2

A = 326,73 cm²

Step-by-step explanation:

The area of a circular cone is area of the base (A₁) plus area lateral (area of a circular sector of radius the slant height)

Then we proceed to calculate the area of the base A₁

diameter of circular base is 8 cm then the radius is 4 cm and the area is:

A₁ = π*r² = 3,14 * (4) ²

A₁ = 3,1416*16 = 50,2656 cm²

Now Lateral area of the cone (A₂) is equal to the area of a circular sector with radius the slant height. We will calculate it, taken into account that this circular sector is part of a circle of radius the slant height.

Between the area of circular sector with radius the slant height and the area of the circle with the same radius there is a linear relation. That is we can calculate area of a circular sector by rule of three as follows:

The length of the circular sector is the length of the circle of the base of the cone, that is:

L = 2*π * (4)

L = 2*3,1416*4

L = 25,1328 cm

Then we have a circular sector of length 25,1328 cm

The area of the circle of radius 22 cm is:

A (c) = π * (22) ² ⇒ A (c) = 1520,5344 cm²

And the length of this circle is:

L (c) = 2*π * (22) ⇒ 138,2304 cm

Then we apply a rule of three

For a length of 138,2304 cm ⇒⇒⇒ (area) 1520,5344 cm²

Then for a length of 25,1328 cm ⇒⇒⇒ (area) A₂ (?)

Therefore:

A₂ = (25,1328) * 1520,5344) / 138,2304

A₂ = 276,4608 cm²

Then total area of the cone is:

A = A₁ + A₂

A = 50,2656 + 276,4608

A = 326,7264 cm²

Round answer A = 326,73 cm²